What is, exactly, a discrete group?

In the setting in which the phrase would be used, $G$ is not simply a group, but a topological group. A discrete group is a topological group in which the topology is discrete.

For example, let us look at the reals under addition, but equip the reals with the discrete topology. This gives us a topological group, which by definition is discrete.

The fact that the reals can be equipped with a non-discrete topology (such as the usual one) which is compatible with addition is not relevant.

"A discrete group is a group equipped with the discrete topology." http://en.wikipedia.org/wiki/Discrete_group

If a set has more than one element then it can be given a non-discrete topology and so it does not make sense to require that "the only topology that can be given is the discrete topology".

The term 'discrete' seems to be applied to a topology here. The unit circle with the euclidean topology is a different topological group from the unit circle with the discretee topology.